WO2025129762A1 - 一种基于二阶滑模控制的航空发动机喘振主动控制系统 - Google Patents

一种基于二阶滑模控制的航空发动机喘振主动控制系统 Download PDF

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WO2025129762A1
WO2025129762A1 PCT/CN2024/070334 CN2024070334W WO2025129762A1 WO 2025129762 A1 WO2025129762 A1 WO 2025129762A1 CN 2024070334 W CN2024070334 W CN 2024070334W WO 2025129762 A1 WO2025129762 A1 WO 2025129762A1
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compressor
sliding mode
control
order
steady
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French (fr)
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孙希明
李世新
全福祥
张志博
杨明绥
李宏新
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Dalian University of Technology
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    • G—PHYSICS
    • G05—CONTROLLING; REGULATING
    • G05B—CONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS
    • G05B13/00—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion
    • G05B13/02—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric
    • G05B13/04—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric involving the use of models or simulators
    • Y—GENERAL TAGGING OF NEW TECHNOLOGICAL DEVELOPMENTS; GENERAL TAGGING OF CROSS-SECTIONAL TECHNOLOGIES SPANNING OVER SEVERAL SECTIONS OF THE IPC; TECHNICAL SUBJECTS COVERED BY FORMER USPC CROSS-REFERENCE ART COLLECTIONS [XRACs] AND DIGESTS
    • Y02—TECHNOLOGIES OR APPLICATIONS FOR MITIGATION OR ADAPTATION AGAINST CLIMATE CHANGE
    • Y02T—CLIMATE CHANGE MITIGATION TECHNOLOGIES RELATED TO TRANSPORTATION
    • Y02T90/00—Enabling technologies or technologies with a potential or indirect contribution to GHG emissions mitigation

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  • the invention belongs to the field of aircraft engine modeling and control, and relates to an aircraft engine surge active control system based on second-order sliding mode control.
  • the traditional engine mainly adopts a control method combining passive anti-surge control and de-surge control.
  • the main idea of this control method is to design the compressor away from the unstable working point so that it has sufficient instability margin, and to take measures to make it exit the instability after the compressor enters the instability.
  • this method is a passive control. Although it can stabilize the engine, it is at the expense of the engine performance, which greatly limits the performance of the compressor and fails to fully exert its potential.
  • the control mechanism of the compressor active stability control theory proposed later is to detect the initial instability disturbance when the instability disturbance is about to occur or at the initial stage, and design a controller to exert active control on the compressor to suppress the development of its internal stall initial disturbance, thereby avoiding the occurrence of instability.
  • the compressor active stability control has many advantages such as improving the engine operating speed and thrust-to-weight ratio, improving its performance at different speeds and different operating states.
  • the compressor active stability control methods can be divided into two categories: modal-based control methods and nonlinear control methods.
  • Modal control is to decompose the compressor stall disturbance signal into multi-order modal waves, detect its amplitude phase, and use the actuator to perform feedback control to prevent the compressor from entering an unstable state.
  • Nonlinear control regards the compressor system as a nonlinear system, and suppresses the compressor stall disturbance through nonlinear control theory, thereby suppressing the occurrence of compressor instability. Therefore, a large number of scholars have combined nonlinear control theory with compressor active control for research, and produced many results.
  • the most widely used nonlinear control methods include inversion control, sliding mode variable structure control, and intelligent control.
  • the compressor system model is assumed to be known.
  • the unmodeled dynamics of the system and internal and external disturbances may make the active control system unable to work properly, so the active control method considering various uncertainties has become a major research hotspot in the active stability control of compressors.
  • sliding mode variable structure control has received widespread attention due to its excellent robustness. It switches between different control actions by adopting the control switching law, thereby generating a "sliding mode" state trajectory.
  • This sliding mode motion is highly robust to parameter perturbations and external disturbances. It is precisely because of the existence of the variable structure form that the sliding mode control method is affected by time lag, spatial lag and system inertia in practical applications.
  • a high-frequency oscillation phenomenon is generated during the control switching process, which is the most prominent chattering problem in sliding mode control. Chattering not only affects the accuracy of control and increases energy consumption, but also the high-frequency unmodeled dynamics in the system are easily excited, destroying the performance of the system, and even causing the system to oscillate or become unstable, damaging the controller components. Therefore, research on chattering suppression is also carried out with the development of sliding mode control theory.
  • the methods are mainly divided into boundary layer method, reaching law method, filtering method, etc.
  • the high-order sliding mode control method not only maintains the advantages of the traditional sliding mode, but also completely solves the "black box" control problem when only the relative order of the system is known. At the same time, it can eliminate the defects of the traditional sliding mode and become a major research hotspot.
  • the present invention proposes an active control system for aircraft engine surge based on second-order sliding mode control.
  • the active control system for aircraft engine surge mainly includes two parts: establishing a compressor model with an actuator and designing a second-order sliding mode controller.
  • the design process of each part includes the following steps:
  • the compressor model is the basis for the design of the active control system for the aircraft engine surge.
  • the compressor model is used to describe the dynamic changes of the pressure ratio, average flow rate and flow disturbance, and provides a basis for the active stability control system to analyze and judge the compressor instability and design the control law.
  • the compressor actuator is the main execution unit of the controller, which affects the pressure rise and flow rate of the compressor by applying active control to stabilize the system.
  • the present invention adopts a first-order spatial Fourier truncated compressor Moore-Gretizer model, and its mathematical model is shown in the figure below:
  • ⁇ is the total static pressure rise coefficient of the compressor system
  • ⁇ is the average flow coefficient
  • ⁇ T represents the average flow coefficient of the throttle valve
  • A is the square of the stall disturbance amplitude, that is, the axial disturbance velocity coefficient, which is used to describe the circumferential asymmetry of the compressor flow. It can reflect the working state and circumferential characteristics of the compressor.
  • B represents the Greitzer-B parameter, which is used to judge the instability state of the compressor
  • l c represents the effective length of the compressor and its upstream and downstream pipelines
  • ⁇ represents the size of the average lag of the compressor stage
  • ⁇ represents the dimensionless time
  • m represents the parameter characterizing the length of the outlet pipeline.
  • the steady-state characteristics of a compressor refer to the steady-state pressure rise characteristics of a compressor without rotating stall and other non-uniform effects. This characteristic is an axisymmetric characteristic of the compressor that is independent of disturbances and can be described by a cubic curve:
  • ⁇ c represents the axisymmetric characteristic of the compressor that is independent of disturbances
  • ⁇ represents the average flow coefficient
  • ⁇ T is the average flow coefficient of the throttle valve, which can be written as:
  • ⁇ T is the throttle valve parameter
  • the first equation shown in formula (1) is the balance equation of the local position of the compressor system
  • the second equation shown in formula (1) is the balance equation of the circumferential average
  • the third equation shown in formula (1) is the mass continuity equation from the cavity to the throttle valve.
  • the throttle valve characteristic line is shown in formula (3), which represents the relationship between the throttle valve pressure rise coefficient and the flow coefficient, as shown in Figure 4;
  • the compressor steady-state characteristic is shown in formula (2), which represents the relationship between the compressor pressure rise coefficient and the flow coefficient in the absence of rotating stall and other non-uniformity effects, as shown in Figure 4.
  • the present invention uses a close-coupled control valve as the actual actuator of the controller.
  • the close-coupled control valve is a valve that is close to the compressor outlet.
  • the function of the valve is the same as that of the compressor, which is to compress air, equivalent to a pure pressure drop at the compressor outlet.
  • the close-coupled control valve By controlling the close-coupled control valve, the pressure rise and flow coefficient of the compressor are changed, thereby affecting the change of momentum and average flow of the compressor flow field, and finally preventing the compressor from entering an unstable state under the action of the control law.
  • the close-coupled control valve is introduced into the compressor model of step 1.1, and finally the compressor system model with a close-coupled control valve is derived as follows:
  • ⁇ v is the pressure drop of the close-coupled valve, which is similar to the throttle valve and can be expressed as the following quadratic curve:
  • ⁇ v is the opening of the close-coupled control valve.
  • the equilibrium point of the compressor is also changed. Since the distance between the close-coupled control valve and the compressor outlet is short enough, the mass storage between the two can be ignored. Therefore, the present invention can regard the compressor and the close-coupled control valve as an equivalent compressor. At this time, the equivalent compressor pressure rise is equivalent to the compressor pressure rise minus the pressure drop of the close-coupled control valve.
  • the equivalent compressor steady-state characteristic line is shown in formula (6), which represents the relationship between the pressure rise coefficient and the flow coefficient of the equivalent compressor, as shown in Figure 4.
  • the system when the system balance point is located on the left side of the maximum value point of the pressure ratio coefficient in the compressor steady-state characteristic line, the system will enter an unstable state under the influence of disturbances; when the system balance point is located on the right side of the maximum value point of the pressure ratio coefficient in the compressor steady-state characteristic curve, the system will always remain stable.
  • the compressor model constructed in step 1.2 after the introduction of the tight-coupled control valve, it can make the compressor steady-state characteristic line move up to the equivalent compressor steady-state characteristic line. At this time, the balance point of the compressor system also becomes the intersection of the throttle valve characteristic line and the equivalent compressor steady-state characteristic line:
  • ⁇ c represents the axisymmetric characteristics of the compressor that are independent of disturbances
  • ⁇ v represents the relationship between the pressure rise coefficient and the flow coefficient of the close-coupled control valve
  • ⁇ cm represents the steady-state characteristics of the equivalent compressor.
  • the disturbances in the compressor are mainly divided into pressure disturbances and flow disturbances. They are also divided into time-varying disturbances and offsets.
  • This patent considers an offset, which is a constant negative mass flow/pressure disturbance that moves the balance of the compression system to the unstable area of the compression diagram, resulting in surge or rotational stall.
  • the pressure offset can be considered as a pressure difference between the compressor and the compressor.
  • the mass flow rate offset can be considered as a certain uncertainty in the throttle valve.
  • d ⁇ represents the external disturbance of the compressor pressure rise
  • d ⁇ represents the external disturbance of the compressor flow
  • u represents the output of the close-coupled control valve, which is also the output of the controller.
  • Sliding mode variable structure control is a control method with strong robustness. It adopts the control switching law and uses the form of variable structure to switch between different control actions, thereby generating a "sliding mode" state trajectory. This sliding mode motion is highly robust to parameter perturbations and external disturbances.
  • Conventional sliding mode variable structure control produces chattering problems due to its variable structure properties, while the high-order sliding mode control method not only maintains the advantages of the invariance of the traditional sliding mode, but also suppresses chattering, eliminates the limitation of relative order and improves control accuracy.
  • S2.1 High-order sliding mode is actually a special type of motion on the integral manifold of a discontinuous dynamic system in the sense of Filippov.
  • s is the sliding manifold, that is, the sliding surface; is the first-order derivative of the sliding surface; is the second-order derivative of the sliding surface; s (r-1) is the r-1-order derivative of the sliding surface; r is the dimension of the constraint condition of the dynamic system.
  • the above formula constitutes the r-dimensional constraint of the dynamic system.
  • Our goal is to design a controller to satisfy this constraint so that the r-dimensional sliding set is non-empty, and it is assumed that it is a local integral set in the sense of Filippov, that is, it consists of the Filippov trajectory of the discontinuous dynamic system. In short, if this equation is satisfied, it is called an r-order sliding mode.
  • the second-order sliding mode control method is the most widely used high-order sliding mode control algorithm because of its simple controller structure and the small amount of information required.
  • the control input explicitly appears in the second-order derivative of the sliding surface.
  • the form is based on s and Or the switching law of their sign functions, to ensure that the state of the system is stable on the sliding surface within a finite time
  • the control law itself is continuous, which effectively suppresses chattering.
  • the four most common algorithms in the second-order sliding mode are the Twisting algorithm, the Sub-Optimal algorithm, the Prescribed Convergence Law algorithm, and the Super-Twisting algorithm.
  • the Super-Twisting algorithm in the second-order sliding mode algorithm adopted by the present invention has the following algorithm form:
  • the Super-Twisting algorithm converges.
  • the phase trajectory of the Super-Twisting algorithm is shown in Figure 5.
  • C is a constant
  • K m is a constant
  • C and K m ensure the finite time stability of the second-order sliding mode control
  • ⁇ is the second-order sliding mode control parameter.
  • ⁇ is the average flow coefficient
  • ⁇ 0 is the target flow value
  • the designed control law is:
  • ⁇ is the intermediate variable of the second-order sliding mode controller
  • ⁇ is the external disturbance of the system
  • ⁇ 1 , ⁇ 2 , ⁇ 3 , ⁇ 4 are constants, representing the parameters of the fast superhelical controller; ⁇ represents the intermediate variable of the controller, which is a variable structure form.
  • the Lyapunov function satisfies the following relationship:
  • coefficients ⁇ min ⁇ AC ⁇ >0, ⁇ min ⁇ B ⁇ >0 are the minimum eigenvalues of matrices AC and B respectively. From formula (20), we can get
  • the system state can converge to the target equilibrium point in a finite time. Therefore, it can be concluded that the designed controller can make the entire compressor system stable.
  • the above is the main design and calculation process of an active control system for aircraft engine surge based on inverse sliding mode control designed by the present invention.
  • the active control system for aircraft engine surge designed by the present invention adopts a second-order sliding mode control method, which not only increases the robustness of the system to non-matching uncertainty, but also weakens the influence of the jitter problem in the sliding mode control.
  • a linear term is added to the super twisting algorithm, which has better convergence characteristics than the ordinary super twisting algorithm; it solves the jitter problem existing in the sliding mode controller, overcomes the compressor surge problem caused by jitter, and expands the effective working range of the active surge controller, thereby realizing the stable operation of the axial flow compressor of the aircraft engine in a wider working range, greatly improving the success rate of active surge control and the stability of the compressor, and improving the safety and reliability of the aircraft engine.
  • FIG1 is a flow chart of the design of an active control system for aircraft engine surge based on second-order sliding mode control
  • FIG2 is a schematic diagram of the structure of an active control system for aircraft engine surge based on second-order sliding mode control
  • FIG3 is a structural diagram of an aircraft engine surge active control system based on second-order sliding mode control in an embodiment of the present invention
  • Figure 5 shows the active control process of surge under undisturbed conditions
  • Figure (a) shows the change process of the local compressor flow coefficient when the fast super-helical second-order controller and the spiral algorithm second-order controller proposed in the present invention implement control under undisturbed conditions
  • Figure (b) shows the change process of the total pressure rise coefficient of the compressor when the fast super-helical second-order controller and the spiral algorithm second-order controller proposed in the present invention implement control under undisturbed conditions
  • Figure (c) shows the change process of the first-order modal amplitude when the fast super-helical second-order controller and the spiral algorithm second-order controller proposed in the present invention implement control under undisturbed conditions.
  • Figure 6 shows the active control process of surge under disturbance conditions, wherein Figure (a) shows the change process of the local compressor flow coefficient when the fast superhelical second-order controller and the spiral algorithm second-order controller proposed in the present invention implement control under white noise disturbance conditions; Figure (b) shows the change process of the total pressure rise coefficient of the compressor when the fast superhelical second-order controller and the spiral algorithm second-order controller proposed in the present invention implement control under white noise disturbance conditions; Figure (c) shows the change process of the first-order modal amplitude when the fast superhelical second-order controller and the spiral algorithm second-order controller proposed in the present invention implement control under white noise disturbance conditions.
  • An active control system for aircraft engine surge based on second-order sliding mode control mainly includes two parts: establishing a compressor model with an actuator and designing a second-order sliding mode controller.
  • the design flow chart of the active control system for aircraft engine surge based on second-order sliding mode control is shown in Figure 1.
  • Figure 2 is a schematic diagram of the structure of an active control system for aircraft engine surge based on backstepping sliding mode control.
  • the controller mainly includes two parts: establishing a compressor model with an actuator and designing a second-order sliding mode controller.
  • FIG3 is a structural diagram of an aircraft engine surge active control system based on second-order sliding mode control in this embodiment.
  • the specific implementation process includes the following steps:
  • the S1 compressor model is the basis for the design of the active control system for the aircraft engine surge.
  • the compressor model is used to describe the dynamic changes of the pressure ratio, average flow rate and flow disturbance, and provides a basis for the active stability control system to analyze and judge the compressor instability and design the control law.
  • the compressor actuator is the main execution unit of the controller, which affects the pressure rise and flow rate of the compressor by applying active control to stabilize the system.
  • the jet device with an independent air source used in the present invention is used as the actuator, and its specific implementation process is as follows:
  • the present invention adopts the first-order spatial Fourier truncated compressor Moore-Gretizer model. This mathematical model As shown below:
  • ⁇ is the average flow coefficient of the compressor system
  • ⁇ is the total static pressure rise coefficient of the compressor system
  • A is the first-order modal amplitude
  • ⁇ T is the average flow coefficient of the throttle valve
  • ⁇ C0 0.30
  • H 0.14
  • W 0.25
  • l C 8.0
  • the equilibrium point of the compressor system is the intersection of the throttle valve characteristic line and the compressor steady-state characteristic line.
  • the actuator used in the present invention is a close-coupled control valve, which is a valve close to the compressor outlet.
  • the function of the valve is the same as that of the compressor, which is to compress air, equivalent to a pure pressure drop at the compressor outlet.
  • ⁇ v is the pressure rise effect brought by the jet device.
  • the equilibrium point of the compressor also changes.
  • the role of the jet device is equivalent to the pure pressure rise at the compressor inlet. Therefore, the compressor and the jet device can be regarded as an equivalent compressor.
  • the equilibrium point of the compressor becomes the intersection of the throttle valve characteristic line and the equivalent compressor steady-state characteristic line.
  • the balance point of the compressor system is the intersection of the throttle valve characteristic line and the equivalent compressor steady-state characteristic line.
  • the system balance point is located on the left side of the maximum point of the pressure ratio coefficient in the compressor steady-state characteristic curve, the system is in an unstable state; otherwise, the system is stable.
  • the balance point of the system is the intersection of the throttle valve characteristic line and the equivalent compressor steady-state characteristic line.
  • the disturbances in the compressor are mainly divided into pressure disturbances and flow disturbances. They are also divided into time-varying disturbances and offsets.
  • the present invention considers an offset, which is a constant negative mass flow/pressure disturbance that moves the balance of the compression system to the unstable area of the compression diagram, resulting in surge or rotational stall.
  • the pressure offset can be regarded as a certain uncertainty in the steady-state characteristics of the compressor; and the mass flow offset can be regarded as a certain uncertainty in the throttle valve.
  • S2 sliding mode variable structure control is a control method with strong robustness. It switches between different control actions by adopting control switching rules to generate a "sliding mode" state trajectory. This sliding mode motion is highly robust to parameter perturbations and external disturbances.
  • Conventional sliding mode variable structure control produces chattering problems due to its variable structure properties, and the high-order sliding mode control method effectively suppresses chattering on the basis of traditional sliding mode, so a high-order sliding mode algorithm is used to design the controller.
  • the specific implementation process is as follows:
  • S2.1 High-order sliding mode is actually a special type of motion on the integral manifold of a discontinuous dynamic system in the sense of Filippov.
  • the Super-Twisting algorithm in the second-order sliding mode algorithm used in this embodiment has the following algorithm form:
  • Figure 5 shows the active control process of surge under undisturbed conditions, wherein Figure 5(a) shows the change process of the local compressor flow coefficient when the fast superhelical second-order controller and the spiral algorithm second-order controller proposed by the present invention implement control under undisturbed conditions; Figure 5(b) shows the change process of the total pressure rise coefficient of the compressor when the fast superhelical second-order controller and the spiral algorithm second-order controller proposed by the present invention implement control under undisturbed conditions; Figure 5(c) shows the change process of the first-order modal amplitude when the fast superhelical second-order controller and the spiral algorithm second-order controller proposed by the present invention implement control under undisturbed conditions. Comparing the fast superhelical second-order controller proposed by the present invention with the spiral algorithm second-order controller, it can be seen from the figure that the fast superhelical second-order controller effectively improves the convergence speed of each state.
  • Figure 6 shows the surge active control process under disturbance conditions, where the meanings of Figure 6(a), Figure 6(b) and Figure 6(c) are the same as those described in Figure 5.
  • the disturbance is a white noise disturbance.
  • the fast super-helical second-order controller No jitter occurred with the spiral algorithm second-order controller, and both controllers can prevent the compressor from entering surge.
  • the designed fast super-helical second-order controller effectively improves the convergence speed of each variable. At the same time, it is less affected by disturbances and has a smaller stability error. Therefore, it can be seen that the designed fast super-helical second-order controller improves the convergence speed and steady-state tracking accuracy of the system, and has stronger anti-interference and robustness.

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Abstract

一种基于二阶滑模控制的航空发动机喘振主动控制系统，包括建立带有执行机构的压气机模型、设计二阶滑模控制器。基于高阶滑模控制的原理，保持传统滑模的强鲁棒性的优点，同时也抑制滑模控制的抖振问题，消除相对阶的限制和提高了控制精度，实现快速、抗干扰能力强、鲁棒性强的喘振主动控制。所设计的快速超螺旋二阶滑模控制器在二阶滑模控制的基础上加入线性项，既有效抑制滑模控制的抖振问题，又增大控制系统的收敛速度，增强系统对非匹配不确定性的鲁棒性；所设计的控制器可以进行有效的主动防喘控制和退喘控制，使压气机能够稳定地工作在喘振边界以外的区域，扩大压气机的工作范围，不需要知道压气机的精确模型；所设计控制器对于未建模动态及系统扰动等不确定性因素具有较强的鲁棒性。

Description

一种基于二阶滑模控制的航空发动机喘振主动控制系统 技术领域
本发明属于航空发动机建模与控制领域，涉及一种基于二阶滑模控制的航空发动机喘振主动控制系统。
背景技术
自上世纪中期以来，随着科技的进步和社会的发展，航空事业得到了蓬勃发展。作为现代航空飞机的动力，航空发动机的性能高低与否，直接决定了飞机的性能及稳定性。航空发动机的发展水平，往往被认为是能够体现国家实力的重要标志。当下，航空发动机在追求更高推重比和更高速度的同时，对其运行稳定性有着更高的要求。压气机作为直接关系其运行状态的主要部件之一，他的稳定性，制约着航空发动机的稳定工作范围。根据压气机的基本原理可知，在追求高推重比和速度的情况下，压气机叶片通道内将产生气流分离，从而导致气动失稳现象。目前通过研究发现航空发动机最主要的两种失稳流态是喘振和旋转失速。不稳定流态的发生，会导致发动机的性能和效率迅速减小，同时不稳定的气流微团产生的振动和温度上升会给发动机结构产生极大的负荷，从极大地缩短了其寿命，增加其维护成本，更严重者甚至会引起发动机故障，产生灾难性后果。因此，为了有效地提高发动机的推重比并扩展其稳定工作范围，避免压气机失稳现象的产生，压气机主动稳定控制技术逐渐成为该领域的研究热点。
为了保证发动机的稳定运行，传统的发动机主要采用被动防喘控制与退喘控制相结合的控制方法，该控制方法的主要思想是设计压气机远离失稳工作点，使其与有足够的失稳裕度，以及在压气机进入失稳后采取措施使其退出失稳。但这种方法是被动控制，虽然能够使发动机稳定，但是以牺牲发动机的性能而换取的，极大限制了压气机的性能，未能完全施展其潜能。而之后提出的压气机主动稳定控制理论的控制机理是在失稳扰动即将产生或出现初期，通过探测失稳初始扰动，设计控制器对压气机施加主动控制以抑制其内部失速初始扰动的发展，从而避免失稳的产生。相比于传统的被动控制方法，压气机主动稳定控制有着提高发动机运行速度和推重比、改善其不同速度以及不同运行状态的性能等诸多优势。目前压气机主动稳定控制方法可以分成两类：基于模态的控制方法和非线性控制方法。模态控制是将压气机失速扰动信号分解为多阶模态波，通过探测其幅值相位，利用执行装置进行反馈控制，进而避免压气机进入失稳状态。非线性控制则是将压气机系统看作一个非线性系统，通过非线性控制理论来抑制压气机的失速扰动，从而抑制压气机失稳状态的发生。因此，大量学者将非线性控制理论与压气机主动控制相结合进行研究，产生了众多成果。目前应用较为广泛的非线性控制方法有反演控制、滑模变结构控制以及智能控制等控制方法。
在大多数主动控制算法中，压气机系统模型都是假设已知的。系统未建模动态及内外部干扰有可能使主动控制系统无法正常工作，所以考虑各种不确定性的主动控制方法成为了压气机主动稳定控制的一大研究热点。而其中滑模变结构控制凭借其以其卓越的鲁棒性能而受到了广泛的重视。它是通过采用控制切换法则在不同的控制作用之间进行切换，从而产生一种“滑动模态”的状态轨迹，这种滑动模态运动对于参数摄动和外界扰动具有强鲁棒性。而也正是变结构形式的存在，使得滑模控制方法在实际应用中受时间滞后、空间滞后以及系统惯性等影响，在控制切换过程中产生的一种高频振荡现象，也就是滑模控制中最突出的抖振问题。抖振不仅影响控制的精确性、增加能量消耗，而且系统中的高频未建模动态很容易被激发起来，破坏系统的性能，甚至使系统产生振荡或失稳，损坏控制器部件。因此，关于抖振抑制的研究也随着滑模控制理论的发展而进行着。方法主要分为边界层法、趋近律法、滤波法等众多方法。其中，高阶滑模控制方法在保持了传统滑模优点的同时，也完全解决了仅仅知道系统相对阶时的一类“黑箱”控制问题，同时能够消除传统滑模的缺陷成为目前一大研究热点。
发明内容
针对现有技术中，压气机主动稳定控制中的抗干扰性差以及滑模变结构控制的抖振问题，本发明提出了一种基于二阶滑模控制的航空发动机喘振主动控制系统。
为了达到上述目的，本发明采用的技术方案为：
一种基于二阶滑模控制的航空发动机喘振主动控制系统，该航空发动机喘振主动控制系统主要包括建立带有执行机构的压气机模型、设计二阶滑模控制器两部分，其各部分设计过程包括以下步骤：
S1建立带有执行机构的压气机模型
压气机模型是该航空发动机喘振主动控制系统设计的基础，压气机模型用于描述压比、平均流量和流量扰动的动态变化，为主动稳定控制系统分析判断压气机失稳和控制律设计提供依据。压气机的执行机构是作为控制器的主要执行单元，通过施加主动控制影响压气机的压升和流量使得系统稳定。其具体实现过程如下：
S1.1本发明所采用的是一阶空间傅里叶截断的压气机Moore-Gretizer模型，其数学模型如下图所示：
其中，Ψ是压气机系统总静压升系数；Φ是平均流量系数，ΦT表示节流阀的平均流量系数；A是失速扰动幅值的平方，也就是轴向扰动速度系数，用于描述压气机流量的周向不对称程度， 可以体现出压气机的工作状态以及周向特性。ΨC0,H,W是压气机稳态特性参数，分别表示Ψ＝0时的压升、压气机稳态特性曲线的半高、压气机稳态特性曲线的半宽。lc,m,α以及B都是压气机结构参数：B表示Greitzer-B参数，用于判断压气机失稳状态；lc表示压气机及其上、下游管道的有效长度；α表示压气机级平均滞后的大小；ξ表示无量纲时间；m表示表征出口管道长度的参数。
压气机稳态特性是指在无旋转失速及其他非均匀性影响情况下压气机的稳态压升特性。该特性是压气机与扰动无关的轴对称特性，可以用三次曲线来描述：
其中，Ψc表示压气机与扰动无关的轴对称特性，ΨC0表示Ψ＝0时的压升；Φ表示平均流量系数。
ΦT是节流阀的平均流量系数，可以写成：
其中，γT是节流阀参数。
通过上述压气机的数学模型可知，公式(1)所示的第一个方程表述为压气机系统局部位置的平衡方程，公式(1)所示的第二个方程表述为周向平均的平衡方程，公式(1)所示的第三个方程则为容腔至节流阀的质量连续方程。同时，根据压气机的基本原理和数学模型可知，当这三个变量的导数为时，压气机系统处于平衡态，此时，该压气机系统的平衡点为节流阀特性线与压气机稳态特性线的交点。所述节流阀特性线为公式(3)所示，表示节流阀压升系数与流量系数之间的关系，由图4中给出；所述压气机稳态特性为公式(2)所示，表示在无旋转失速及其他非均匀性影响情况下压气机的压升系数与流量系数之间的关系，由图4给出。
S1.2本发明采用紧联控制阀作为控制器的实际执行装置，紧联控制阀是一个紧贴压气机出口的阀门，该阀门的作用与压气机的作用一样都是压缩空气，相当于压气机出口处的一个纯压降，通过控制紧联控制阀从而改变压气机的压升和流量系数，进而影响压气机流场的动量和平均流量的变化，最终在控制律的作用下避免压气机进入失稳状态。然后将紧联控制阀引入步骤1.1的压气机模型中，最终推导出带有紧联控制阀的压气机系统模型如下：
其中，Ψv为紧连阀的压降，与节流阀相似，可以表示为如下二次曲线：
其中，γv为紧联控制阀开度。
根据步骤1.1中对于压气机系统平衡态的描述，在引入紧联控制阀后，压气机的平衡点也因此发生改变。由于紧联控制阀与压气机出口间的距离足够短，使得两者之间的质量储存可以忽略不计。所以本发明可以将压气机与紧联控制阀看成一个等效压气机，此时，等效压气机压升等价于压气机压升减去紧联控制阀的压降。等效压气机的稳态特性Ψcm(Φ)可以用压气机稳态特性Ψc(Φ)与紧联控制阀的特性Ψv(Φ)表示：
Ψcm(Φ)＝Ψc(Φ)-Ψv(Φ)       (6)
此时，压气机的平衡点变为节流阀特性线与等效压气机稳态特性线的交点。所述等效压气机稳态特性线如公式(6)所示，表示等效压气机的压升系数与流量系数之间的关系，由图4中给出。
S1.3根据步骤1.2所构建的压气机模型可知，在未引入执行机构的情况下，压气机系统的平衡点为节流阀特性线与等效压气机稳态特性线的交点：
而根据压气机的基本原理可知，当系统平衡点位于压气机稳态特性线中压比系数极大值点的左侧部分时，系统会在扰动的影响下进入失稳状态；当系统平衡点位于压气机稳态特性曲线中压比系数极大值点的右侧部分时，系统会始终保持稳定状态。而根据步骤1.2所构建的压气机模型可知，在引入紧联控制阀后，它可以使得压气机稳态特性线上移变为等效压气机稳态特性线，此时压气机系统的平衡点也变为节流阀特性线与等效压气机稳态特性线的交点：
其中，Ψc表示压气机与扰动无关的轴对称特性；Ψv表示紧联控制阀的压升系数与流量系数之间的关系特性；Ψcm表示等效压气机的稳态特性。
图4中展示了不同紧联控制阀开度下的压气机特性。从图中可以看出，紧联控制阀的引入使得系统平衡点保持在等效压气机稳态特性线极大值点的右边，由此也可以看出紧联控制阀可以使得步骤1.2所构建的压气机系统稳定。
S1.4与其他物理系统一样，扰动将发生在压缩系统中。扰动的存在可能会影响压气机的稳定性，甚至可能会导致控制器不能正常工作。因此在设计主动控制器时，扰动的抑制也应在考虑范围内。压气机中的扰动主要分压力扰动和流量扰动。同时也分为时变扰动和偏移量。本专利考虑的是一种偏移量，偏移量是一个恒定的负质量流量/压力扰动使得压缩系统的平衡移到压缩图的不稳定区域，导致产生喘振或者旋转失速。其中压力偏移量可被认为是压气机 稳态特性中某种不确定性；而质量流量偏移量可被认为是节流阀中的某种不确定性。在引入这些扰动情况下，公式(4)所示的压气机系统模型变为：
其中，dψ表示压气机的压升外部扰动；dφ表示压气机的流量外部扰动；u表示紧联控制阀输出，也是控制器输出。
S2设计二阶滑模控制器
滑模变结构控制是一种具有较强鲁棒性的控制方法，它是通过采用控制切换法则，利用变结构的形式在不同的控制作用之间进行切换，从而产生一种“滑动模态”的状态轨迹，这种滑动模态运动对于参数摄动和外界扰动具有强鲁棒性。常规的滑模变结构控制因其变结构性质而产生抖振问题，而高阶滑模控制方法既保持了传统滑模的不变性的优点，同时也抑制了抖振，消除了相对阶的限制和提高了控制精度。
S2.1高阶滑模实际上是在Filippov意义下，不连续动态系统的一种特殊类型的积分流形上的运动。
其中，s为滑动流形，也就是滑模面；为滑模面的一阶导数；为滑模面的二阶导数；s(r-1)为滑模面的r-1阶导数；r为该动态系统的约束条件的维数.
也就是说通过上述式子构成该动态系统的r维约束条件，我们的目的是设计控制器满足这个约束条件，使得该r维滑动集非空，且假设他是Filippov意义下局部积分集，就是说它由不连续动态系统的Filippov轨迹组成。总之，若满足该等式，则称之为r阶滑模。
S2.2二阶滑模
二阶滑模因为控制器结构简单且所需要的信息不多，成为目前应用最广泛的高阶滑模控制算法。在二阶滑模控制中方法，控制输入显式地出现在滑模面的二阶导数中，控制律的形式是基于s和或它们的符号函数的切换规律，保证系统的状态在有限时间内稳定于滑模面上，但控制律本身却是连续的，从而有效的抑制了抖振。二阶滑模中最为常见的四种算法为Twisting(螺旋)算法、Sub-Optimal(次优)算法、Prescribed Convergence Law(给定收敛律)算法和Super-Twisting(超螺旋)算法。本发明所采用的二阶滑模算法中的Super-Twisting(超螺旋)算法，其算法形式如下：
其中，u为控制器输出；λ为常数，为二阶滑模控制参数；s为滑模面；u1为该二阶滑模控制器 中间变量，为变结构形式；β为常数，为变结构控制参数；为中间变量的导数；
当满足条件：
则Super-Twisting(超螺旋)算法收敛。Super-Twisting(超螺旋)算法相轨迹如图5所示。其中，C为常数；Km为常数，C和Km保证二阶滑模控制的有限时间稳定性；λ为二阶滑模控制参数。
S2.3二阶滑模控制器设计
S2.3.1针对上述系统(9)，将其分成流量子系统和压升子系统。首先针对流量子系统设计滑模面s1：
s1＝Φ-Φ0          (13)
其中，Φ为平均流量系数；Φ0为目标流量值；
对滑模面求导可得：
继续对滑模面求二阶导数：
设计控制律为：
其中，σ为二阶滑模控制器中间变量；Δ为系统外部扰动；
然后，在超螺旋算法中增加线性项来提高控制器的收敛速度：
S2.3.2证明该控制律的稳定性，定义Lyapunov函数为：
V＝2η3|s|+η4s2+0.5σ2+0.5(η1|s|0.5sgn(s)+η2s-σ)2      (18)
其中，η1、η2、η3、η4为常数，表示快速超螺旋控制器参数；σ表示该控制器中间变量，为变结构形式。
写成如下二次型形式：
Lyapunov函数满足如下关系式:
其中，||Γ||2为Γ的二范数；λmin{Q},λmax{Q}分别为矩阵Q的最小特征值和最大特征值,可知λmin{Q}>0,λmax{Q}>0.对Lyapunov函数求导可得：
其中有如下形式：
由式子(21)和(22)可得：
如果ΓT(A-C)Γ和ΓTBΓ都为正定二次型,则其充分必要条件是矩阵A-C和B所有的顺序主子式都大于0,由推导可得
当η1,η2,η3,η4为常数且取值满足式(24)时,矩阵A-C和B的特征值都大于0,由式子(23)可得
其中:系数λmin{A-C}>0,λmin{B}>0分别为矩阵A-C和B的最小特征值.由式(20)可得
由式子(20)，(25)和(26)可得：
由于式(27)的参数ξ1和ξ2都大于0,可知系统状态在有限时间内收敛到s＝0，σ＝0，由表达式(17)可知受扰快速super twisting算法在有限时间内收敛到且收敛时间满足
因此，系统状态可在有限时间收敛到目标平衡点。所以得出，所设计的控制器可以使得整个压气机系统的稳定性。
由式(27)可知,当Lyapunov函数距离平衡点较近时，非线性项ξ1V0.5远大于线性项ξ2V,收敛速度主要由非线性项ξ1V0.5决定，其Lipschitz性质使得系统的收敛速度很快。当Lyapunov函数距离平衡点较远时，线性项ξ2V远大于非线性项ξ1V0.5，收敛速度主要由线性项ξ2V决定，为指数收敛。非线性项与线性项的结合，使本文的快速super twisting算法具有很快的收敛速度。
以上即为本发明所设计的一种基于反演滑模控制的航空发动机喘振主动控制系统的主要设计及计算过程。通过本发明设计的航空发动机喘振主动控制系统，该方法采用二阶滑模控制方法，既增加系统对非匹配不确定性的鲁棒性，又削弱了滑模控制中抖振问题的影响，然后在使用超螺旋算法中增加线性项，具有比普通super twisting算法更优良的收敛特性；解决了滑模控制器中存在的抖振问题，克服了抖振引起的压气机喘振问题，扩大了喘振主动控制器的有效工作范围，从而实现航空发动机轴流压气机在更宽的工作范围内的稳定工作，很大程度上提高了喘振主动控制的成功率和压气机的稳定性，提高了航空发动机的安全性和可靠性。
附图说明
图1为基于二阶滑模控制的航空发动机喘振主动控制系统设计流程图；
图2为基于二阶滑模控制的航空发动机喘振主动控制系统结构示意图；
图3为本发明的某实施例中基于二阶滑模控制的航空发动机喘振主动控制系统结构图；
图4为引入紧联控制阀后不同喷气流量系数下压气机特性示意图，其中实线表示压气机未引入紧联控制阀的稳态特性曲线；点划线表示压气机引入紧联控制阀后，且紧联控制阀开度γv＝1.6时压气机的等效稳态特性曲线；虚线表示压气机引入紧联控制阀后，且紧联控制阀开度γv＝2.16时压气机的等效稳态特性曲线；两条点线表示节流阀开度为γT＝0.7和γT＝0.62时节流阀特性曲线。
图5为无扰动情况下喘振主动控制过程，其中图(a)为本发明所提出的快速超螺旋二阶控制器与螺旋算法二阶控制器在无扰动情况下实施控制时，局部压气机流量系数的变化过程；图(b)为本发明所提出的快速超螺旋二阶控制器与螺旋算法二阶控制器在无扰动情况下实施控制时，压气机总压升系数的变化过程；图(c)为本发明所提出的快速超螺旋二阶控制器与螺旋算法二阶控制器在无扰动情况下实施控制时，一阶模态幅值的变化过程。
图6为有扰动情况下喘振主动控制过程，其中图(a)为本发明所提出的快速超螺旋二阶控制器与螺旋算法二阶控制器在有白噪声扰动情况下实施控制时，局部压气机流量系数的变化过程；图(b)为本发明所提出的快速超螺旋二阶控制器与螺旋算法二阶控制器在有白噪声扰动情况下实施控制时，压气机总压升系数的变化过程；图(c)为本发明所提出的快速超螺旋二阶控制器与螺旋算法二阶控制器在有白噪声扰动情况下实施控制时，一阶模态幅值的变化过程。
具体实施方式
下面结合附图及本发明实施例，对本发明内容进行进一步说明。
一种基于二阶滑模控制的航空发动机喘振主动控制系统，该控制系统主要包括建立带有执行机构的压气机模型、设计二阶滑模控制器两部分，基于二阶滑模控制的航空发动机喘振主动控制系统的设计流程图如图1所示。
图2为基于反演滑模控制的航空发动机喘振主动控制系统的结构示意图。从图中可以看出，控制器主要包含建立带有执行机构的压气机模型、设计二阶滑模控制器两部分。
图3为本实施例中基于二阶滑模控制的航空发动机喘振主动控制系统的结构图。
其具体实施过程包括以下步骤：
S1压气机模型是该航空发动机喘振主动控制系统设计的基础，压气机模型用于描述压比、平均流量和流量扰动的动态变化，为主动稳定控制系统分析判断压气机失稳和控制律设计提供依据。压气机的执行机构是作为控制器的主要执行单元，通过施加主动控制影响压气机的压升和流量使得系统稳定。本发明所采用的带独立气源的喷气装置作为执行机构，其具体实现过程如下：
S1.1本发明所采用的是一阶空间傅里叶截断的压气机Moore-Gretizer模型，该数学模型 如下所示：
其中，Φ为压气机系统的平均流量系数，Ψ为压气机系统的总静压升系数，A为一阶模态幅值，ΦT为节流阀的平均流量系数，方程中其他参数均为压气机固有参数，在此选取以下数值：ΨC0＝0.30，H＝0.14，W＝0.25，lC＝8.0，α＝1/3.5，m＝1.75。
根据压气机的基本原理和数学模型可知，当这三个变量的导数为时，系统处于平衡态，此时，该压气机系统的平衡点为节流阀特性线与压气机稳态特性线的交点。
S1.2本发明所采用的执行机构是紧联控制阀，它是一个紧贴压气机出口的阀门，该阀门的作用与压气机的作用一样都是压缩空气，相当于压气机出口处的一个纯压降，通过控制紧联控制阀从而改变压气机的压升和流量系数，进而影响压气机流场的动量和平均流量的变化，最终在控制律的作用下避免压气机进入失稳状态。因此将其引入的步骤1.1的压气机模型，推导可得带紧联控制阀的压气机系统模型如下：
其中Ψv为喷气装置带来的压升效应。
根据步骤1.1中对于压气机系统平衡态的描述，在引入喷气装置后，压气机的平衡点也因此发生改变。此时，喷气装置的作用相当于压气机进口处的纯压升，因此，可以将压气机与喷气装置看作为一个等效的压气机。此时，压气机的平衡点变为节流阀特性线与等效压气机稳态特性线的交点。
S1.3根据压气机的基本原理和数学模型可知，在未引入执行机构的情况下，压气机系统的平衡点为节流阀特性线与等效压气机稳态特性线的交点。当系统平衡点位于压气机稳态特性曲线中压比系数极大值点的左侧部分时，系统处于失稳状态；否则，系统稳定。系统的平衡点为节流阀特性线与等效压气机稳态特性线的交点。当系统平衡点交于压气机稳态特性曲线压比系数极大值点的左边，则系统进入失稳状态，否则，系统稳定。在引入紧联控制阀后，它可以使得压气机稳态特性线上移变为等效压气机稳态特性线，此时压气机系统的平衡点也变为节流阀特性线与等效压气机稳态特性线的交点。图4为不同紧联控制阀开度下的压气机特性。从图中可以看出，紧联控制阀的引入使得系统平衡点保持在等效压气机稳态特性线极大值点的右边，从而得出紧联控制阀可以使得步骤1.2所构建的压气机系统稳定。
S1.4与其他物理系统一样，扰动将发生在压缩系统中。压气机中的扰动主要分压力扰动和流量扰动。同时也分为时变扰动和偏移量。本发明考虑的是一种偏移量，偏移量是一个恒定的负质量流量/压力扰动使得压缩系统的平衡移到压缩图的不稳定区域，导致产生喘振或者旋转失速。其中压力偏移量可被认为是压气机稳态特性中某种不确定性；而质量流量偏移量可被认为是节流阀中的某种不确定性。
S2滑模变结构控制是一种具有较强鲁棒性的控制方法，它是通过采用控制切换法则在不同的控制作用之间进行切换，从而产生一种“滑动模态”的状态轨迹，这种滑动模态运动对于参数摄动和外界扰动具有强鲁棒性。常规的滑模变结构控制因其变结构性质而产生抖振问题，而高阶滑模控制方法在传统滑模的基础上有效抑制了抖振，所以采用高阶滑模算法设计控制器。具体实现过程如下：
S2.1高阶滑模实际上是在Filippov意义下，不连续动态系统的一种特殊类型的积分流形上的运动。高阶滑模控制器设计目的就是满足r维滑动集非空，也称之为为r阶滑模。当r＝2时，为二阶滑模。本实施例采用的二阶滑模算法中的Super-Twisting(超螺旋)算法，其算法形式如下：
然后针对流量误差设计滑模面，并设计快速Super-Twisting控制律，也就是在其基础上增加线性项来提高控制器的收敛速度，算法形式如下：
其中具体参数如下：
η1＝10,η2＝10,η3＝20,η4＝5         (33)
本实施案例的仿真计算结果如图5、图6所示：图5为无扰动情况下的喘振主动控制过程，其中图5(a)为本发明所提出的快速超螺旋二阶控制器与螺旋算法二阶控制器在无扰动情况下实施控制时，局部压气机流量系数的变化过程；图5(b)为本发明所提出的快速超螺旋二阶控制器与螺旋算法二阶控制器在无扰动情况下实施控制时，压气机总压升系数的变化过程；图5(c)为本发明所提出的快速超螺旋二阶控制器与螺旋算法二阶控制器在无扰动情况下实施控制时，一阶模态幅值的变化过程。将本发明所提出的快速超螺旋二阶控制器与螺旋算法二阶控制器做对比，从图中可以看出，该快速超螺旋二阶控制器有效提高了各个状态的收敛速度。
图6为有扰动情况下的喘振主动控制过程，其中图6(a)、图6(b)以及图6(c)的意义与图5中所描述的相同。该扰动是一个白噪声扰动，在该扰动的影响下，快速超螺旋二阶控制器 与螺旋算法二阶控制器均未发生抖振，两个控制器都可以避免压气机进入喘振，相比于螺旋算法二阶控制器，所设计的快速超螺旋二阶控制器有效提高了各个变量的收敛速度，同时，受扰动影响更小，稳定误差更小，因此可以看出所设计的快速超螺旋二阶控制器提高了系统的收敛速度和稳态跟踪精度，具有更强的抗干扰性和鲁棒性。
以上所述实施例仅表达本发明的实施方式，但并不能因此而理解为对本发明专利的范围的限制，应当指出，对于本领域的技术人员来说，在不脱离本发明构思的前提下，还可以做出若干变形和改进，这些均属于本发明的保护范围。

Claims (2)

  1. 一种基于二阶滑模控制的航空发动机喘振主动控制系统，其特征在于，所述的航空发动机喘振主动控制系统包括建立带有执行机构的压气机模型、设计二阶滑模控制器两部分，其各部分设计过程包括以下步骤：
    步骤S1建立带有执行机构的压气机模型；
    步骤S2设计二阶滑模控制器。
  2. 根据权利要求1所述的一种基于二阶滑模控制的航空发动机喘振主动控制系统，其特征在于，所述的步骤S1、步骤S2具体如下：
    步骤S1建立带有执行机构的压气机模型，具体如下：
    S1.1采用一阶空间傅里叶截断的压气机Moore-Gretizer模型，其模型如下所示：
    其中，Ψ是压气机系统总静压升系数；Φ是平均流量系数，ΦT表示节流阀的平均流量系数；A是失速扰动幅值的平方，也就是轴向扰动速度系数，用于描述压气机流量的周向不对称程度，可以体现出压气机的工作状态以及周向特性；ΨC0,H,W是压气机稳态特性参数，分别表示Ψ＝0时的压升、压气机稳态特性曲线的半高、压气机稳态特性曲线的半宽；lc,m,α以及B都是压气机结构参数：B表示Greitzer-B参数，用于判断压气机失稳状态；lc表示压气机及其上、下游管道的有效长度；α表示压气机级平均滞后的大小；ξ表示无量纲时间；m表示表征出口管道长度的参数；
    压气机稳态特性是压气机与扰动无关的轴对称特性，采用三次曲线进行描述：
    其中，Ψc表示压气机与扰动无关的轴对称特性，ΨC0表示Ψ＝0时的压升；Φ表示平均流量系数；
    ΦT是节流阀的平均流量系数，可以写成：
    其中，γT是节流阀参数；
    通过上述压气机模型可知，公式(1)中第一个方程表述为压气机系统局部位置的平衡方程，公式(1)中第二个方程表述为周向平均的平衡方程，公式(1)中第三个方程则为容腔至节流阀的质量连续方程；同时，当三个变量的导数为时，压气机系统处于平衡态，此时，该压气机系统的平衡点为节流阀特性线与压气机稳态特性线的交点；所述节流阀特性线为公式(3)所示，表示节流阀压升系数与流量系数之间的关系；所述压气机稳态 特性为公式(2)所示，表示在无旋转失速及其他非均匀性影响情况下压气机的压升系数与流量系数之间的关系；
    S1.2采用紧联控制阀作为控制器的实际执行装置，通过控制紧联控制阀改变压气机的压升和流量系数，进而影响压气机流场的动量和平均流量的变化，最终在控制律的作用下避免压气机进入失稳状态；将紧联控制阀引入步骤S 1.1的压气机模型中，最终推导出带有紧联控制阀的压气机系统模型如下：
    其中，Ψv为紧连阀的压降，与节流阀相似，可以表示为如下二次曲线：
    其中，γv为紧联控制阀开度；
    根据步骤1.1中对于压气机系统平衡态的描述，在引入紧联控制阀后，压气机的平衡点也因此发生改变；将压气机与紧联控制阀看成一个等效压气机，此时，等效压气机压升等价于压气机压升减去紧联控制阀的压降；等效压气机的稳态特性Ψcm(Φ)用压气机稳态特性Ψc(Φ)与紧联控制阀的特性Ψv(Φ)表示：
    Ψcm(Φ)＝Ψc(Φ)-Ψv(Φ)      (6)
    此时，压气机的平衡点变为节流阀特性线与等效压气机稳态特性线的交点；所述等效压气机稳态特性线如公式(6)所示，表示等效压气机的压升系数与流量系数之间的关系；
    S1.3根据步骤1.2所构建的压气机模型可知，在未引入执行机构的情况下，压气机系统的平衡点为节流阀特性线与等效压气机稳态特性线的交点：
    根据步骤1.2所构建的压气机模型可知，在引入紧联控制阀后，使得压气机稳态特性线上移变为等效压气机稳态特性线，此时压气机系统的平衡点也变为节流阀特性线与等效压气机稳态特性线的交点：
    其中，Ψc表示压气机与扰动无关的轴对称特性；Ψv表示紧联控制阀的压升系数与流量系数之间的关系特性；Ψcm表示等效压气机的稳态特性；
    S1.4在引入这些扰动情况下，公式(4)所示的压气机系统模型变为：
    其中，dψ表示压气机的压升外部扰动；dφ表示压气机的流量外部扰动；u表示紧联控制阀输出，也是控制器输出；
    S2设计二阶滑模控制器，具体如下：
    S2.1通过公式(10)构成该动态系统的r维约束条件，设计控制器满足该约束条件，则称之为r阶滑模；
    其中，s为滑动流形，也就是滑模面；为滑模面的一阶导数；为滑模面的二阶导数；s(r-1)为滑模面的r-1阶导数；r为该动态系统的约束条件的维数；
    S2.2二阶滑模
    采用二阶滑模算法中的Super-Twisting超螺旋算法，其算法形式如下：
    其中，u为控制器输出；λ为常数，为二阶滑模控制参数；s为滑模面；u1为该二阶滑模控制器中间变量，为变结构形式；β为常数，为变结构控制参数；为中间变量的导数；
    当满足条件：
    则Super-Twisting算法收敛；
    其中，C为常数；Km为常数，C和Km保证二阶滑模控制的有限时间稳定性；λ为二阶滑模控制参数；
    S2.3二阶滑模控制器设计
    S2.3.1针对公式(9)所示的压气机系统模型，将其分成流量子系统和压升子系统；首先针对流量子系统设计滑模面s1：
    s1＝Φ-Φ0         (13)
    其中，Φ为平均流量系数；Φ0为目标流量值；
    依次对滑模面求导、对滑模面求二阶求导后，设计控制律为：
    其中，σ为二阶滑模控制器中间变量；Δ为系统外部扰动；
    在超螺旋算法中增加线性项来提高控制器的收敛速度：
    S2.3.2通过验证控制律的稳定性，系统状态可在有限时间收敛到目标平衡点，所设计的控制器可以使整个压气机系统的稳定性。
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